Many of the museums are run with income from foundations and largely private resources, I believe.

The Tate, for instance, was founded by Sir Henry Tate, who paid for the building and donated his collection. According to the Tate website: "Each year, the majority of our funding must come from private sources. We depend on the support of individuals, trusts, foundations and corporations to help us continue to care for and build our Collection, and engage with our audiences."

Many of the museums have fun or attractive "containers" for donations by the doors and these always seem to fill nicely. Wonderful system. :)

The British Museum, which is also free, is a national museum and I suspect as such it receives taxpayer funding.